We examine the quantum Hall (QH) states of the optical lattices with square geometry using Bose–Hubbard model (BHM) in presence of artificial gauge field. In particular, we focus on the QH states for the flux value of \(\alpha = 1/3\). For this, we use cluster Gutzwiller mean field (CGMF) theory with cluster sizes of \(3\times 2\) and \(3\times 3\). We obtain QH states at fillings \(\nu = 1/2, 1, 3/2, 2, 5/2\) with the cluster size \(3\times 2\) and \(\nu = 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3\) with \(3\times 3\) cluster. Our results show that the geometry of the QH states is sensitive to the cluster sizes. For all the values of \(\nu \), the competing superfluid (SF) state is the ground state and QH state is the metastable state.

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